Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Calculate the derivative
![The equation given is:
\[ y = \sqrt[4]{\frac{2x}{4x - 3}} \]
This equation represents a function \( y \) defined as the fourth root of the fraction \(\frac{2x}{4x - 3}\).
### Explanation:
- **Numerator**: The numerator of the fraction is \(2x\), a simple linear expression indicating that as \(x\) increases, the numerator increases proportionally.
- **Denominator**: The denominator is \(4x - 3\), another linear expression. For values of \(x\) that make this expression zero (i.e., \(x = \frac{3}{4}\)), the function is undefined, as division by zero is not possible.
- **Fourth Root**: The entire fraction is within the expression for a fourth root. This means that once the linear expressions are evaluated, you take the fourth root of the result.
This function implies that the domain excludes values where the denominator is zero and possibly other constraints depending on the behavior of the fourth root for negative values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84aa1b0e-781e-4210-8d80-8938e43d2409%2Fde4a5c09-3e68-494d-9cba-a4e432d053b8%2F2d831il_processed.png&w=3840&q=75)
Transcribed Image Text:The equation given is:
\[ y = \sqrt[4]{\frac{2x}{4x - 3}} \]
This equation represents a function \( y \) defined as the fourth root of the fraction \(\frac{2x}{4x - 3}\).
### Explanation:
- **Numerator**: The numerator of the fraction is \(2x\), a simple linear expression indicating that as \(x\) increases, the numerator increases proportionally.
- **Denominator**: The denominator is \(4x - 3\), another linear expression. For values of \(x\) that make this expression zero (i.e., \(x = \frac{3}{4}\)), the function is undefined, as division by zero is not possible.
- **Fourth Root**: The entire fraction is within the expression for a fourth root. This means that once the linear expressions are evaluated, you take the fourth root of the result.
This function implies that the domain excludes values where the denominator is zero and possibly other constraints depending on the behavior of the fourth root for negative values.
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